Very often we make use of the identity \(\partial _x^2\Lambda ^{-2}=\Lambda ^{-2}-1\)  whenever such operator appears and is defined. For \(s\ge t\) , then \(H^s(X)\hookrightarrow H^{t}(X)\) , and \(H^s(X)\)  is an algebra as soon as \(s>1/2\) , that is, if \(u,v\in H^s(X)\) , then \(uv\in H^s(X)\) . Additionally, given \(u,v\in H^s(X)\) , with \(s>1/2\) , then \(\Vert uv\Vert _{H^{s-1}(X)}\le c\Vert u\Vert _{H^s(X)}\Vert v\Vert _{H^{s-1}(X)}\) , for some constant \(c>0\)  depending on \(s\) . For further details, see [1]} and [2]}.
